#48:
Electromagnetic Theory and Computation: A Topological Approach
Up: MSRI Publications
MSRI Publications -- Volume 48
Electromagnetic Theory and Computation: A Topological Approach
By Paul W. Gross and P. Robert Kotiuga
Although topology was recognized by Gauss and Maxwell to play a
pivotal role in the formulation of electromagnetic boundary value
problems, it is a largely unexploited tool for field computation. The
development of algebraic topology since Maxwell provides a framework
for linking data structures, algorithms, and computation to
topological aspects of three-dimensional electromagnetic boundary value
problems. This book attempts to expose the link between Maxwell and a
modern approach to algorithms.
The first chapters lay out the relevant facts about homology and
cohomology, stressing
their interpretations in electromagnetism. These topological
structures are subsequently tied to variational formulations in
electromagnetics, the finite element method, algorithms, and certain
aspects of numerical linear algebra. A recurring theme is the
formulation of and algorithms for the problem of making branch cuts
for computing magnetic scalar potentials and eddy currents. An appendix
bridges the gap between the material presented and standard expositions
of differential forms, Hodge decompositions, and tools for realizing
representatives of homology classes as embedded manifolds.